Three point charges lie in a straight line along the y-axis. A charge of q1 = -9.10 µC is at y = 6.70 m, and a charge of q2 = -7.70 µC is at y = -3.60 m. The net electric force on the third point charge is zero. Where is this charge located?
Let q3 be the value of the third point charge which is located at y = x. We will apply the principle of superposition to find the location where net electric force on point charge q3 is zero.
The force on q3 due to q1 is given by Coulomb's Law:
F13 = k * |q1 * q3| / (x - 6.70)^2
Similarly, the force due to q2 on q3 is given by:
F23 = k * |q2 * q3| / (x + 3.60)^2
Since the net force on charge q3 is zero, these two forces must have the same magnitude but opposite direction. Therefore, we can equate the expressions for F13 and F23:
k * |q1 * q3| / (x - 6.70)^2 = k * |q2 * q3| / (x + 3.60)^2
Now, we can cancel out the k * |q3| from both sides, and rearrange the equation:
-9.10 / (x - 6.70)^2 = -7.70 / (x + 3.60)^2
Cross-multiplying and taking the square root, we get:
sqrt(-9.10*(x + 3.60)^2)=sqrt(-7.70*(x - 6.70)^2)
Now, squaring both sides we get:
-9.10*(x + 3.60)^2=-7.70*(x - 6.70)^2
Now, divide both sides by -1 to remove the negative sign:
9.10*(x + 3.60)^2=7.70*(x - 6.70)^2
And then, divide both sides by 7.70 and 9.10:
(x + 3.60)^2 / 7.70 = (x - 6.70)^2 / 9.10
Now, replace x by y:
(y + 3.60)^2 / 7.70 = (y - 6.70)^2 / 9.10
As this is a quadratic equation after this simplification, we get two solutions:
y = -3.21 m and y = 1.53 m
However, since charge q2 is already at a location y = -3.60 m, the point charge q3 cannot be located at y = -3.21 m. Therefore, the third charge must be located at:
y = 1.53 m.
To find the location of the third point charge where the net electric force is zero, we can use the principle of superposition and the formula for electric force between two charges.
The formula for electric force between two charges is given by:
F = k * |q1 * q2| / r^2
Where:
F is the electric force between two charges
k is the electrostatic constant (9 x 10^9 Nm^2/C^2)
q1 and q2 are the magnitudes of the charges
r is the distance between the charges
In this case, we have two charges, q1 and q2, located at y1 = 6.70 m and y2 = -3.60 m respectively.
Let's assume that the third charge, q3, is located at y3.
Since the net electric force on the third charge is zero, the net force due to the first and second charges must be equal in magnitude but opposite in direction.
We can write this as:
F1 = -F2
Using the formula for electric force, we have:
k * |q1 * q3| / (y3 - y1)^2 = k * |q2 * q3| / (y3 - y2)^2
Here, we can cancel out the electrostatic constant (k) and convert the magnitudes of the charges (q1, q2, q3) to their absolute values to simplify the equations.
|q1 * q3| / (y3 - y1)^2 = |q2 * q3| / (y3 - y2)^2
Substituting the given values:
q1 = -9.10 µC
q2 = -7.70 µC
y1 = 6.70 m
y2 = -3.60 m
And solving for y3, we can find the location of the third charge where the net electric force is zero.
To find the location of the third point charge where the net electric force is zero, we can apply the principle of superposition, which states that the net force on a charge due to multiple charges is the vector sum of the individual forces due to each charge.
In this scenario, we have two point charges, q1 and q2, and the third charge, q3, for which we want to determine the location. Let's assume q3 has a charge of q3 = q.
Given:
q1 = -9.10 µC
y1 = 6.70 m
q2 = -7.70 µC
y2 = -3.60 m
Using Coulomb's Law, the electric force between two point charges q₁ and q₂ can be calculated as follows:
F = (k * q₁ * q₂) / r²
Where k is the electrostatic constant equal to k = 8.99 x 10^9 Nm²/C², and r is the distance between the two charges.
To find the location of the third charge where the net electric force is zero, we can set up an equation:
F₁ + F₂ = 0
Where F₁ is the force exerted by q₁ on q₃, and F₂ is the force exerted by q₂ on q₃.
Let's calculate these forces:
F₁ = (k * q * q₁) / r₁²
F₂ = (k * q * q₂) / r₂²
Since the charges q₁, q₂, and q₃ lie along the y-axis, the distances r₁ and r₂ can be calculated as the differences in their y-coordinates:
r₁ = |y - y₁|
r₂ = |y - y₂|
We can substitute these values into the equation and solve for y.
(k * q * q₁) / r₁² + (k * q * q₂) / r₂² = 0
Let's plug in the given values:
(k * q * (-9.10 µC)) / (|y - 6.70 m|)² + (k * q * (-7.70 µC)) / (|y + 3.60 m|)² = 0
Now we can solve this equation for y.